Problem
GEO-B1-M06-P023 Product of Areas With Intersecting Diagonals
#23
★★★★☆ Level 4 of 5
In convex quadrilateral \(ABCD\), the diagonals meet at point \(O\). Prove that \(S_{AOB}\cdot S_{COD}=S_{BOC}\cdot S_{DOA}\).
Write each of the four triangle areas using two sides and the sine of the angle between the diagonals.
Let the angle between the diagonals be \(\varphi\). Then \(S_{AOB}=\frac{1}{2}AO\cdot BO\sin\varphi\), \(S_{COD}=\frac{1}{2}CO\cdot DO\sin\varphi\). Similarly, \(S_{BOC}=\frac{1}{2}BO\cdot CO\sin\varphi\), \(S_{DOA}=\frac{1}{2}DO\cdot AO\sin\varphi\). The products on the left and right are identical, so the equality is proved.
A strong but very useful formula for quadrilaterals.